DOI QR코드

DOI QR Code

REGULAR BRANCHED COVERING SPACES AND CHAOTIC MAPS ON THE RIEMANN SPHERE

Lee, Joo-Sung

  • Published : 2004.07.01

Abstract

Let (2,2,2,2) be ramification indices for the Riemann sphere. It is well known that the regular branched covering map corresponding to this, is the Weierstrass P function. Lattes [7] gives a rational function R(z)= ${\frac{z^4+{\frac{1}{2}}g2^{z}^2+{\frac{1}{16}}g{\frac{2}{2}}$ which is chaotic on ${\bar{C}}$ and is induced by the Weierstrass P function and the linear map L(z) = 2z on complex plane C. It is also known that there exist regular branched covering maps from $T^2$ onto ${\bar{C}}$ if and only if the ramification indices are (2,2,2,2), (2,4,4), (2,3,6) and (3,3,3), by the Riemann-Hurwitz formula. In this paper we will construct regular branched covering maps corresponding to the ramification indices (2,4,4), (2,3,6) and (3,3,3), as well as chaotic maps induced by these regular branched covering maps.

Keywords

chaotic map;branched covering space;Weierstrass P function;the Riemann sphere

References

  1. Math. Tidsskrift v.B On Normal Subgroups with finite Index in F-Groups S. Bundgaard;J. Nielsen
  2. An Introduction to Chaotic Dynamical Systems R. Devaney
  3. Acta Math. v.171 A proof of Thurston's Topological Characterization of Rational Functions A. Douady;J. H. Hubbard https://doi.org/10.1007/BF02392534
  4. London Math. Soc. Lecture Notes Series 9 Elliptic Functions and Elliptic Curves P. Du Val
  5. Math. Tidsskrift v.B On Fenchel's Conjecture about F-Groups R. Fox
  6. The Functions of Mathematical Physics H. Hochstadt
  7. CR Acad. Sci. Paris v.166 Sur l'iterationdes substitutions rationalles et les fonctions de Poincare S. Lattes
  8. Dynamics in One Complex Variable J. Milnor
  9. Pitman Research Notes in Math. Series 161 Branched Coverings and Algebraic Functions M. Namba
  10. Transl. Math. Monogr. Elliptic Functions and Elliptic Integrals V. Prasolov;Y. Solovyev